Matchmaking is the process of matching two people together or two sets of people together, for example in the context of sporting events or in online gaming. Such a process may be of particular importance in numerous contexts. For the sake of illustration, the selection of a gamer to compete with another gamer should be carefully made. If the game level of the selected gamer is too low, the other gamer will not enjoy playing the game and, on the contrary, if the game level of the selected gamer is too high, the other gamer can be frustrated.
Such processes may also be used for preselecting items as a result of selecting a first item, for example to suggest goods to be bought online after having selected first goods. They are particularly useful when a huge amount of items can be selected and when the selection mechanisms are implemented in devices such as smartphones having small displays.
A well-known matchmaking process is based on the Bradley-Terry model that uses a score assigned to each individual or each team. According to this model, it is possible to predict the outcome of a comparison of items based on a score associated with each of these items. For the sake of illustration, the probability that an individual i wins a game against an individual j, a score Si being assigned to individuals i and j, respectively, can be defined as follows:
      P    ⁡          (              i        >        j            )        =            S      i                      S        i            +              S        j            
The score denoted Sk that represents the skill of individual k can be determined based on the number of times individual k has won a match.
While the Bradley-Terry model is used for gaming it can also be used in numerous applications such as applications for ranking the relevance of documents in a searching process.
There exist a lot of processes for computing skills of individuals. Among these processes, the ELO rating system is a method for determining the relative skill levels of players such as chess players. The difference of skills between players is used to predict match results. According to the ELO rating system, the winner takes points from the loser after each match. The difference between the ratings of the winner and loser determines the number of points to be won and lost so that the rating system is self-correcting.
However, although these matchmaking algorithms may have been proven to be useful in numerous applications, they have drawbacks resulting from their transitive property according to which if an individual i is ranked above an individual j and if an individual j is ranked above an individual k, then individual i is ranked above an individual k.
Since there are many applications in which it is requested to select at least one item from among a set of items based on scores associated with the items, there is a need for nontransitive matchmaking processes and nontransitive ranking processes.